Assuming a ship of 500 m in length, and weighing 10,000,000 tons (100 times heavier than the largest aircraft carrier), the firing of a 20kg shell at 1.3% of the speed of light requires a momentum transfer of 0.008 m/s from the ship. Assuming constant acceleration, the firing requires a shell acceleration of 1.5E10 g, which is done in 130 microseconds.

A velocity change of 0.008 m/s in 130 microseconds means the ship itself has an average acceleration of about 60m/s/s or about 6 g. However, this assumes the entire ship is rigidly coupled to the firing system. If held together by normal structural forces, which propagate at best at multiples of the speed of sound, then no more than 1/2 meter of other structure can be involved. It's more likely the equipment itself will have 100s of g of stress.

The acceleration requires 1.5E14 Joules, or 35 kT of TNT, which is what the clip said.